Questions on the mathematical aspects of signal processing. Please consider first if your question might be more suitable for http://dsp.stackexchange.com/

learn more… | top users | synonyms (1)

1
vote
0answers
13 views

How to create obtain aliased version of $f(t)$ by upsampling whenever $f(t)$ at every $t$ is available

Suppose there is original complex-valued $f(t)$ with $t$ ranging from $-\infty$ to $\infty$. It is possible obtain samples from original $f(t)$ at every $t$ with some negligible error. If one ...
1
vote
0answers
28 views

Eigenvalues for correlation matrix which have the form of an harmonic function

As a continuation to this question, I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\frac{|\phi_1-\phi_2|}{2}} &...
0
votes
1answer
11 views

Unit of the second derivative of the power spectral density

To characterize a subtle oscillation embedded in a time varying voltage signal measured in microvolts, I took the second derivative of the PSD (which I computed as the fft of the autocorrelation) ...
2
votes
0answers
28 views

Is it possible to use regularization to minimize the (expected) number of non-zero digits in a number?

This question may be slightly related to this question on length of the representation of a number in a certain basis. Introduction / Background In image and video coding, particularly the ...
0
votes
0answers
7 views

How do I validate my ARMAX model?

Say I have some ouput $y_1, y_2, \ldots, y_N$ and inputs $x_1, x_2, \ldots, x_N$ which, by various time series methods, I've found to match an ARMAX(2,2,1) model. So I've found the estimations for ...
0
votes
0answers
23 views

Variance of $\hat{b} =\underset{b}{\mathrm{argmin}} \sum_{t=1}^N [y_t - bu_t]^2 $

We have $y_t = bu_t + e_t$ where $u_t$ is the input signal and I'm trying to find an expression for the variance of the estimate for b that is determined to be $$\hat{b} = \underset{b}{\mathrm{argmin}...
5
votes
2answers
116 views

How can I recover a sequence of numbers given a corrupted version of it?

I have an unknown sequence of real numbers $x_i$ and a known sequence of real numbers $y_i$; $y_i$ is a corrupted version of $x_i$, i.e., $$y_i=x_i+n_i$$ where $n_i$ is a random number distributed ...
1
vote
0answers
41 views

What is the output $y(t)$ when you have input $x(t) = \cos(2 \pi t) $ and frequency response response $h(t) = u(t) - u(t - 1/2)$?

The output $y(t)$ is the convolution of input $x(t)$ with impulse response $h(t)$: $$ y(t) = h(t) * x(t) $$ This is a linear, time invariant system. What is the output $y(t)$ in real form when you ...
1
vote
0answers
80 views

Solving a non-linear parametric equation

I am interested in solving a parametric equation where the unknown function is a function of time, and there is also an input. For example: $ y^{2}(t) + y(t) = \sin(t)$ I am coming from a signal ...
1
vote
0answers
20 views

Single Sideband LSB-SC Demodulation

The problem is how the phase φ effects the outcome when the input(message signal) is the DSB-SC LSB. It's : message: $m(t)=A_{m}cos(ω_{m}t)$ carrier: $c(t)=A_{c}cos(w_{c}t)$ I found that the LSB ...
0
votes
0answers
35 views

Can downsampling create energy at the Nyquist frequency?

I am a bit surprised by the following and would like to share it with you. I expect I am mistaken somewhere and will be happy to be corrected. I have searched StackExchange not only in Mathematics ...
1
vote
0answers
23 views

How to find out the Power of $x(t)$?

I am studying signals and system. I learned that \begin{align} P&=\lim_{L\to\infty} \frac 1{2L} \int_{-L}^{L} |x(t)|^2 dt\\ P&=\frac 1{T} \int_{<T>} |x(t)|^2 dt ~~~\mbox{, P could be ...
1
vote
1answer
24 views

In signals processing why is the discrete sequence x[n] undefined (as opposed to 0) when n is not an integer?

In Oppenheim & Schafer's "Discrete Time Signals Processing" it's written that: ... it is important to recognize that x[n] is defined only for integer values of n. It is not correct to think of x[...
1
vote
0answers
28 views

Why is this defined as $u[n]$?

In LTI systems there are two famous functions the unit step function and the unit impulse functions. And they are defined as follows. $$ u[n] = \begin{cases} 1, & \text{if $n$ $\ge$ 0} \\ 0, &...
0
votes
0answers
18 views

Is it possible to do effectively irrational-interval sampling of a continuous signal?

Suppose there is a real-valued $f(t)$, with $t$ being time. And one wishes to sample at interval of $\pi$, for example. Perfect irrational-interval sampling is not possible, but is there a way to do ...
0
votes
1answer
33 views

is there any good way to figure out number of fourier series frequencies of some signal?

Suppose you have $f(t)$, but you do not know the exact function and can only measure $f(t)$ at certain time. Assume $f(t)$ is complex-valued with $t$ being "time." One wishes to find out the number ...
1
vote
1answer
18 views

Relationship between short-time and large-frequency asymptotics in Fourier transform

I am trying to understand how the short-time behaviour of a function $f(t)$ influences the large-frequency asymptotics of its Fourier transform $g(\omega)=\mathcal{F}[f(t)](\omega)\equiv \tilde{f}(\...
4
votes
0answers
47 views

Signal processing and algebraic geometry

Signal processing is a pretty huge branch of what I would (maybe wrongly) call electrical engineering. I have heard here and there whispers of interesting connections between signal processing - in ...
0
votes
0answers
20 views

How to sketch frequency response obtained from H(z)?

How to sketch frequency response obtained from H(z)? I'm adding an example and its solution below. I did not understand some of the things for question 7.4, part D. Any help is appreciated! There ...
2
votes
1answer
24 views

Convolution Problem

while working on a signal processing problem i've reached to the following: So my aproach was: Am I doing something wrong? Is it valid Y(f)=[X(f) x H(f)]*W(f)=X(f) x [H(f)*W(f)] If you could ...
1
vote
0answers
37 views

Fast evaluation of an integral convolution with an “expanding kernel”

Suppose I have a 1-D integral convolution transform like this: $$ g(x) = \int_{-\infty}^{+\infty} dy\, f(y)\, K(x-y). \qquad (1) $$ Say the kernel $K(x)$ is a known analytic function, and say we have ...
0
votes
1answer
25 views

Find the inverse z transform of $H(z)=\frac{1}{8-6z^{-1}+z^{-2}}$

This question was on a homework assignment, and the solutions have been distributed but I'm having trouble reproducing the solutions. Given the initial conditions $y[-1]=y[-2]=0$ and the difference ...
0
votes
1answer
16 views

The set of inputs x(n) to a system is described with a small superscript T - what is that?

I think it means TRANSPOSE, but I can't figure out the need to perform a transpose operation: Similarly, the set of weights that go with the inputs is written with a small T as well: There is ...
1
vote
1answer
22 views

Graph of the Angle of a Fourier Transform

If I need to graph the magnitude and angle of a discrete Fourier transform which happens to be $X(e^{j\omega}) = 4\cos(4\omega)$, I know how to graph the magnitude, but how do you graph the angle? I ...
1
vote
1answer
40 views

A signal on a noisy channel is input to a filter

Question: A Wide-sense stationary (weakly stationary, (WSS)) random signal {X(t)}t∈R with Power spectral density(PSD) S_X(ω) is transmitted on a noisy channel where it is disturbed by an additive ...
-1
votes
1answer
31 views

Periodic product of sinusoids

(This is problem P-3.7 from the book 'Signal processing first') Let $x(t) = 2\cos(\omega_1t)\cos(\omega_2t) = \cos([\omega_1 + \omega2]t)+\cos([\omega_2 - \omega_1]t)$ where $0 < \omega_1 < \...
0
votes
1answer
35 views

Is this system time-invariant?

I think this system is not time-invariant, but I'm not really sure how to plug in a couple test cases to check. The system is: $x(t)$ -->(S)--> $y(t) = \int_{-\infty}^{3t}x(\tau) d\tau$ Without ...
2
votes
0answers
59 views

Source estimation for identification of anomalous events

I’m stuck on the following problem. There are two sources $S_A$ and $S_B$ at the ends of a channel. Both are made up of a white noise component $W_i$ plus an impulsive component $I_i$: $S_A = W_A + ...
1
vote
1answer
25 views

How to find period of a sum of periodic functions

I got this function: $$ x[n]=\sin(2*\pi*4/3*n) + \cos(2*\pi*5/2*n) $$ It is easy to see that period of the sin is 3/4 and the ...
0
votes
0answers
35 views

Deriving difference equation from a rational system function $H(z)$

If I have the system function $H(z)$ of a linear time-invariant system, how do I derive the difference equation relating its input $x(n)$ and output $y(n)$? The system function is given by $$H(z) = \...
2
votes
0answers
319 views

Intuition behind the DTFT vs Fourier transform of ideally sampled signal

So I am taking a signal processing course in EE and my professor is an Engineer who reallly likes math however his book which we use for the class falls in the dreadfull purgatory of math books in my ...
1
vote
0answers
39 views

Magnitude and Angle of Discrete Fourier Transform

I can't figure out how to get the magnitudes for periodic discrete Fourier transforms. For example if $x[n] = cos(\frac{\pi}{4}n + \frac{\pi}{2})$, I need to find and plot the magnitude $|X(e^{jw})|$...
1
vote
0answers
41 views

Why is the cross-correlation an integral?

The cross-correlation of continuous $f,g$ is: $$(f \star g)(\tau)=\int_{-\infty}^{\infty}f^*(t)g(t+\tau)dt$$ Why is it an integral? Why doesn't The cross-correlation of continuous $f,g$ is: $$(f \...
0
votes
0answers
24 views

Why does cross-correlation involve the complex conjugate?

The cross-correlation of continuous $f,g$ is: $(f \star g)(\tau)=\int_{-\infty}^{\infty}f^*(t)g(t+\tau)dt$ where * is the complex conjugate. Why is there the complex conjugate?
1
vote
0answers
14 views

Complex filter factorizations with invariant points

Based on this question, using the same $z_0$: $$z_0 = e^{2\pi i / 8}$$ if we modify the sequence from previous question to look like this ($*$ denotes discrete convolution): $$\left(z_0^{[-2k,3k]} * ...
1
vote
0answers
18 views

Complex filter factorizations - continued

Continuing from this rather silly trivial question factoring real valued filters into shorter complex ones, hoping this won't be as trivial. If we modify it a bit: $$z_0 = e^{2\pi i / 8}$$ and $$\...
1
vote
0answers
23 views

Complex filter factorizations

There is a famous low pass filter $[1,2,1]$ in signal processing which can be factored in the sense of a convolution product over the real numbers : $[1,1] * [1,1]$. This is the only way to do it over ...
1
vote
1answer
70 views

Signal processing : future values prediction

Let $f : \mathbb{R}^+ \rightarrow \mathbb{R} $ be a continuous function. Do you have some references (books or online resource) about techniques that allow to predict $f(x_{n+1})$, knowing $f(x_0), .....
1
vote
1answer
21 views

Second Order Damped System (Harmonic Oscillator)

I'm currently working on an electrical engineering problem involving an ideal operational amplifier. The system itself is governed by the transfer function, which relates output to input: $$\omega_o^...
2
votes
0answers
37 views

Extracting a Cosine Function from a Linear Combination of Cosines

I have a frequency modulated signal which must contain only $ g(t)=B.\cos(\omega(t).t+\phi)$, but it gets the form as below $$ f(t)=B.\cos(\omega(t).t+\phi)+A_1.\cos(\omega(t-t_1).t+\phi_1)+A_2.\cos(\...
0
votes
1answer
46 views

Inverse $Z$ transform of $\frac{1}{z-a}$

I don't really get what's happening here and I haven't been able to find a single example on how to get the inverse $Z$-transform of $\frac{1}{z-a}$. Can anyone show the way?
0
votes
1answer
50 views

is a “non-causal” system “memory”?

Is their a relationship between non-causality and memory? for example: is the system $Y(t) = X(t+1)$ memory or memory-less. I got confused because the memory system is defined to depend only on the (...
2
votes
1answer
82 views

Calculating convolution integral analytically

How can i compute convolution integral analytically, without using graphs. I hate using graphs, shiftings which are error prone. If this is possible can you explain what way i must follow? For ...
2
votes
1answer
44 views

How to mathematically model noise?

In my project I have to perform analysis of noise effect in certain signal. I am just wondering how is noise formally described? Up to now I always simulate a noisy signal using MATLAB in an additive ...
1
vote
0answers
38 views

Relation between Covariance and Energy of a random signal

Let's say I have the below random signal: $Y[n]=[y(n)y(n−1)y(n−2)....y(1)]$ I have two random variables now: The first one $X_1$ which express the maximum eigenvalue of the covariance matrix of $Y$. ...
1
vote
0answers
15 views

Does inverse of all Fouriers transforms have a corresponding function in time domain?

I am trying to cancel out the following transfer function of a system: $$\frac{( 1 - e^{(i*k*T)} ) }{ (i*k)}$$ I thought it would work if I find the inverse Fourier transform of $$\frac{ (i*k)}{( 1 ...
1
vote
2answers
65 views

Why is $\int e^{-t}u(t) dt = (1-e^{-t})u(t) + Constant$?

How do you solve $\int e^{-t}u(t) dt $? In which u(t) is the unit step function. $\int e^{-t}u(t) dt = (1-e^{-t})u(t) + Constant$ But what are the intermediate steps? Unit step u(t) = \begin{...
0
votes
0answers
22 views

explanation of correlation of stationary stochastic processes

I have some questions about correlation in stationary stochastic processes. I know that the expectation of a random variable is $E(x)=\int_{-\infty}^{+\infty} a ...
0
votes
3answers
64 views

Why is the convolution output in terms of 't' not $\tau$?

The convolution integral is defined as: $$y(t) = (h * x)(t) = \int^{+\infty}_{-\infty} h(\tau). x(t-\tau)\ d\tau$$ where $h(t)$ and $x(t)$ are functions in terms of time. Why is $y$ in terms of '$t$...
1
vote
2answers
43 views

Do decaying exponential signals have finite energy?

"A signal that decays exponentially has finite energy, so, it is also an energy signal." http://www.songho.ca/dsp/signal/signals.html I don't quite get how that can be true. Energy of a ...